Stacey has a $8,000 balance on her credit card that has an interest rate of 21%,
compounded monthly.
(a) If she decides to pay it off over 5 years with equal monthly payments, how much should
each payment be?
(b) How much interest will Stacey pay to the credit card company?
(c) If instead she wants to completely pay off her debt after 3 years (i.e. 2 years early),
what lump sum payment must she make?
a) Well, Stacy's credit card balance is like the Energizer Bunny, it just keeps going and going! To calculate the equal monthly payments to pay off her $8,000 debt in 5 years with a 21% interest rate compounded monthly, you'll need to put on your math hat. Let's break it down:
First, calculate the monthly interest rate by dividing the annual interest rate by 12: 21% / 12 = 1.75% per month.
Now, we need to figure out the total number of months in 5 years: 5 years * 12 months/year = 60 months.
To determine the equal monthly payment, we can use the formula for monthly payment on an amortizing loan:
Payment = (balance * monthly interest rate) / (1 - (1 + monthly interest rate) ^ -number of months)
Plugging in the numbers, Stacy's monthly payment would be:
Payment = (8000 * 0.0175) / (1 - (1 + 0.0175) ^ -60)
Warning! Math overload! *Beep boop*
b) Now, let's calculate the amount of interest Stacy will pay over the 5-year period. To find this, we just need to subtract the original balance of $8,000 from the total amount paid (equal monthly payment * number of months).
c) If Stacy wants to pay off her debt after only 3 years, she'll have to make a lump sum payment to get rid of it early. To find this amount, we can use the same formula as before but adjust the number of months:
Lump sum payment = (balance * (1 + monthly interest rate) ^ -number of months)
Plugging in the numbers, we have:
Lump sum payment = 8000 * (1 + 0.0175) ^ -36
Remember to include this information in your note to Stacy: Dear Stacy, love the enthusiasm to get rid of your debt early! Just wanted to mention that making a big payment now will save you from paying a lot of interest in the long run! *Wink*
To calculate the answers, we can use the formula for calculating the equal monthly installment payment of a loan:
Monthly Payment = [Principal * (Interest Rate * (1 + Interest Rate)^Number of Payments)] / [(1 + Interest Rate)^Number of Payments - 1]
Let's calculate the answers step-by-step:
(a) Calculate the monthly payment amount if Stacey decides to pay off the balance over 5 years (60 months) with equal monthly payments.
Monthly Payment = [8,000 * (0.21/12 * (1 + 0.21/12)^60)] / [(1 + 0.21/12)^60 - 1]
Calculating this equation will give us the amount of each monthly payment.
(b) Calculate the total interest paid over 5 years.
Total Interest Paid = (Monthly Payment * Number of Payments) - Principal
(c) Calculate the lump sum payment required to completely pay off the debt after 3 years (36 months).
Lump Sum Payment = [Principal * (1 + Interest Rate * (1 + Interest Rate)^Number of Payments)] / [(1 + Interest Rate)^Number of Payments - 1]
Let's calculate each step:
(a) Calculate the monthly payment amount:
Monthly Payment = [8,000 * (0.21/12 * (1 + 0.21/12)^60)] / [(1 + 0.21/12)^60 - 1]
Monthly Payment ≈ $207.97
Each monthly payment should be approximately $207.97.
(b) Calculate the total interest paid over 5 years:
Total Interest Paid = ($207.97 * 60) - $8,000
Total Interest Paid ≈ $5,077.95
Stacey will pay approximately $5,077.95 in interest over the 5 years.
(c) Calculate the lump sum payment required to completely pay off the debt after 3 years:
Lump Sum Payment = [8,000 * (1 + 0.21/12 * (1 + 0.21/12)^36)] / [(1 + 0.21/12)^36 - 1]
Lump Sum Payment ≈ $4,141.20
To completely pay off the debt after 3 years, Stacey will need to make a lump sum payment of approximately $4,141.20.
To find out the answers to these questions, we need to use the formula for calculating the monthly payment on a loan with compound interest. The formula is:
𝐏 = 𝑃 × (𝑟 × (1 + 𝑟)^𝑛) / ((1 + 𝑟)^𝑛 − 1)
Where:
𝑃 = Principal balance (initial loan amount)
𝑟 = Monthly interest rate (annual interest rate / 12)
𝑛 = Total number of payments
(a) To find the monthly payment for Stacey's credit card balance over 5 years, we substitute the given values into the formula:
𝑃 = $8,000
𝑟 = 21% / 12 = 0.0175 (monthly interest rate)
𝑛 = 5 years × 12 = 60 (total number of payments)
Using the formula, the monthly payment (𝐏) can be calculated as:
𝐏 = $8,000 × (0.0175 × (1 + 0.0175)^60) / ((1 + 0.0175)^60 - 1)
(b) To find the total interest Stacey will pay to the credit card company, we need to subtract the principal amount from the total amount repaid. The total amount repaid is equal to the monthly payment multiplied by the number of payments. So:
Total amount repaid = Monthly payment × Total number of payments
Total interest paid = Total amount repaid - Principal amount
(c) To find the lump sum payment if Stacey wants to completely pay off her debt after 3 years, we need to find the present value of the remaining loan balance. We can use the formula for the present value of a future value on a loan:
𝑃𝑉 = 𝐹𝑉 / (1 + 𝑟)^𝑛
Where:
𝑃𝑉 = Present value
𝐹𝑉 = Future value (remaining balance after 3 years)
𝑟 = Monthly interest rate (annual interest rate / 12)
𝑛 = Total number of payments remaining (2 years × 12 months = 24 payments remaining)
By substituting the values into the formula, we can calculate the lump sum payment (𝑃𝑉).
.21/12 = .0175 (monthly rate)
5 years = 60 months
8000 = p(1 - 1.0175^-60 )/.0175
p = 216.43
b) what do you think?
c) lump payment = 8000(1.0175)^36 - 216.43(1.0175^36 - 1)/.0175
= ....